Geometrically motivated hyperbolic coordinate conditions for numerical relativity: Analysis, issues and implementations

TL;DR

This paper investigates hyperbolic coordinate conditions in numerical relativity motivated by geometry, analyzing their properties, challenges at excision boundaries, and proposing alternatives based on Killing's equation.

Contribution

It introduces geometrically motivated hyperbolic coordinate conditions, analyzes their implications for excision techniques, and explores alternative conditions derived from Killing's equation.

Findings

Common coordinate conditions lead to non-outflow modes at excision boundaries.

Boundary conditions are needed for certain modes, complicating simulations.

Alternative conditions based on Killing's equation show promising numerical results.

Abstract

We study the implications of adopting hyperbolic driver coordinate conditions motivated by geometrical considerations. In particular, conditions that minimize the rate of change of the metric variables. We analyze the properties of the resulting system of equations and their effect when implementing excision techniques. We find that commonly used coordinate conditions lead to a characteristic structure at the excision surface where some modes are not of outflow-type with respect to any excision boundary chosen inside the horizon. Thus, boundary conditions are required for these modes. Unfortunately, the specification of these conditions is a delicate issue as the outflow modes involve both gauge and main variables. As an alternative to these driver equations, we examine conditions derived from extremizing a scalar constructed from Killing's equation and present specific numerical examples.